where \(G_m(\theta _k)=\frac{1}{m}\sum _{i=1}^m \nabla q(z_i^{(k)},\theta _k)\)  is an unbiased estimate of \(G_n(\theta _k)\)  based on \(m<n\)  randomly chosen observations \(z_1^{(k)},\dots ,z_m^{(k)}\)  drawn at each \(k\) . Setting \(e_k = G_m(\theta _k) - G_n(\theta _k)\)  results in the same setting as stochastic approximation above. Though \(m=1\)  is computationally inexpensive and is a popular choice, the stochastic gradients \(G_m(\theta _k)\)  are very noisy and a small \(\gamma _k\)  satisfying (REF ) is needed to compensate. This results in slow convergence making it well suited for problems where the gradients \(G_m\)  are very cheap to computed and running very many iterations is feasible. Unfortunately, this is typically not the case for more complicated economic models. A number of methods can accelerate convergence,Improvements to sgd  include momentum [1]} and accelerated gradient-descent [2]}. As for gd, using \(P_k=I_d\)  in sgd  can lead to very slow convergence. Popular extensions of stochastic gradient descent include adagrad [3]}, and adam [4]}. but they tend to converge slower than nr, or bfgs. More closely related to this paper, some study stochastic bfgs, but compute the full sample \(G_n\)  many times which is more demanding.See e.g. Algorithm 1, step 4, in [5]}. Note that stochastic Newton-Raphson (snr), with a small \(m =1\) , is not used in practice because \(H_m^{(k)}\)  is often not invertible when \(m<d_\theta \) .
